Modeling and identification of uncertain-input systems

TL;DR

This paper introduces uncertain-input models for system identification, leveraging Gaussian processes and Bayesian methods to estimate unknown inputs and system responses, with practical algorithms and applications to classical problems.

Contribution

It presents a novel framework for uncertain-input system identification using Gaussian processes, with new algorithms for hyperparameter estimation and approximations for intractable cases.

Findings

Effective estimation of unknown inputs and system responses.

Unified approach encompassing classical system identification problems.

Numerical simulations demonstrating the model's applicability.

Abstract

In this work, we present a new class of models, called uncertain-input models, that allows us to treat system-identification problems in which a linear system is subject to a partially unknown input signal. To encode prior information about the input or the linear system, we use Gaussian-process models. We estimate the model from data using the empirical Bayes approach: the input and the impulse responses of the linear system are estimated using the posterior means of the Gaussian-process models given the data, and the hyperparameters that characterize the Gaussian-process models are estimated from the marginal likelihood of the data. We propose an iterative algorithm to find the hyperparameters that relies on the EM method and results in simple update steps. In the most general formulation, neither the marginal likelihood nor the posterior distribution of the unknowns is tractable. Therefore, we propose two approximation approaches, one based on Markov-chain Monte Carlo techniques and one based on variational Bayes approximation. We also show special model structures for which the distributions are treatable exactly. Through numerical simulations, we study the application of the uncertain-input model to the identification of Hammerstein systems and cascaded linear systems. As part of the contribution of the paper, we show that this model structure encompasses many classical problems in system identification such as classical PEM, Hammerstein models, errors-in-variables problems, blind system identification, and cascaded linear systems. This allows us to build a systematic procedure to apply the algorithms proposed in this work to a wide class of classical problems.